In a joint paper Artur Avila and myself proved (the easy part!) that almost all translation flows are weakly mixing. The proof was based on non-vanishing a Lyapunov exponents of the so-called Kontsevich---Zorich cocycle and on a “linear elimination” procedure. In this talk we introduce a new cohomological cocycle (over the KZ cocycle), based on twisted cohomology, with the goal of proving “effective weak mixing”: lower bounds on the local dimension of spectral measures, polynomial speed of convergence of Cesaro averages of correlations, speed of ergodicity of product flows (with circle rotations) for smooth functions. This work was motivated by results of Bufetov and Solomyak in special cases (substitutions systems and genus two translation flows) by different, but analogous, methods.